Now imagine an experiment where initially the system starts in the
space \(\mathcal {S}\)  and begins to evolve unitarily. We keep making
measurements, at regular intervals of time \(\tau \) , to find if the system
has arrived in the detector subspace \(\mathcal {D}\) . If the result of the measurement is negative, the system (now projected back into \(\mathcal {S}\) ) continues its unitary evolution, until the next measurement
and the process is repeated. The experiment stops when we get a positive
result indicating arrival into \(\mathcal {D}\) . Thus the evolution
of the state vector consists of a sequence of steps in \(\mathcal {S}\) , each consisting
of a unitary evolution followed by the projection \(Q\) . Looking at
the state of the system conditioned on survival (non-detection), let
us denote by \(\left|\psi (n\tau )\right\rangle \)  the state just after the
\(n\) -th measurement. Then it can be shown [1]} that
\(\left|\psi (n\tau )\right\rangle =\widetilde{U}_{\tau }^{n} \left|\psi (0)\right\rangle , \quad \widetilde{U}_{\tau }=Q U_{\tau } Q.\) 
